Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 08/Solution 1

This question is testing implicit differentiation, since $y$ is a function of $x$. We start by taking logarithm of both sides of equation to get $$\begin{array}{r}\log (y)=\log (x^{x^4})=x^4\log x\end{array}$$

Then differentiating with respect to $x$, we apply chain rule to $\log (y)$, and product rule to $=x^4\log x$: $$\begin{array}{rl}\frac{y^{\prime }}{y}& =4x^3\log x+\frac{x^4}{x}\\ & =4x^3\log x+x^3.\end{array}$$

Multiplying both sides by $y$ we get: $y^{\prime }=(4x^3\log x+\frac{x^4}{x})y$, where $y=x^{x^4}$. Thus the final answer is $$\begin{array}{rl}{y}^{\prime }& =(4x^3\log x+\frac{x^4}{x})x^{x^4}.\end{array}$$